Let $X_1,X_2,\dots,X_n$ be i.i.d. random vectors in $\mathbb{R}^d$ having mean $\mu$ and finite covariance matrix $V_0$ of rank $q>0$. Let $$S=\frac1n\sum_{i=1}^n(X_i-\mu)(X_i-\mu)^\prime,$$ then we have $$n(\bar{X}-\mu)^\prime S^{-1}(\bar{X}-\mu)\overset{d}\to\chi^2_q.$$
First question: Is this statement true? What's the name of this theorem, and where can I find this theorem or how can I prove it?
Note that we can express $EX_i=\mu$ as $E(X_i-\mu)=0$.Let $X_1,X_2,\dots,X_n$ be i.i.d. random vectors in $\mathbb{R}^d$. Suppose $E[m(X_i,\mu)]=0$, where $m(X_i,\mu)\in\mathbb{R}^s$ and $Var(m(X_i,\mu))$ is finite with rank $q>0$. Let $$M=\frac1n\sum_{i=1}^nm(X_i,\mu)$$ and $$Q=\frac1n\sum_{i=1}^nm(X_i,\mu)m^\prime(X_i,\mu)$$ Can we conclude that $$nM^\prime Q^{-1}M\overset{d}\to\chi^2_q?$$
Second question: Is this generalized form of the first statement true? What's the name of this theorem, and where can I find this theorem or how can I prove it? Do we need further assumptions to function $m(X_i,\mu)$?